# Why time dilation is enlarged over long distance ? (small movement of head leads to huge shift of the plane of "present")

Considering the special relativity, we know that the time for the moving object (relative to us, for example) is slower (relative to us).

Now consider the Andromeda paradox explained by Roger Penrose (Ref. https://en.wikipedia.org/wiki/Rietdijk%E2%80%93Putnam_argument), why is this tiny different of speed between two persons (and accordingly tiny different of time dilation) is enlarged over distance?

The essence of the question in the title deals with the "relativity-of-simultaneity". (My reply will focus on that, because the Andromeda paradox and Rietdijk–Putnam argument are not needed to answer your question.)

In short, the relativity of simultaneity implies that
"the set of events at time=t according to BLUE (along the x-axis)"
are different from
"the set of events at time=t' according to RED (along the x'-axis)"
as drawn on a spacetime diagram.

Consider the triangles $$OQ_1P_1$$ and $$OQ_2P_2$$.
By similarity (proportionality), $$\frac{Q_2P_2}{OP_2}=\frac{Q_1P_1}{OP_1}.$$ So, the further away the "galaxy" is from where the people meet at event O, the larger difference in time between the corresponding events at the galaxy P ("now" according to RED) and Q ("now" according to BLUE).

(UPDATE: I didn't see @MarcoOcram 's answer while I was composing my answer. That answer has the essence of what I have in my answer.)

By the way, the diagram in Wikipedia page you linked has the wrong orientation for the "black car's T axis" if it is supposed to be a spacetime diagram.

• Thank you, I think I was confused between time dilation and relativity of simultaneity. Because when you check the time dilation equation, there is nothing related to the distance between the observers, but I understood now (Pls correct me if wrong) that it is all about the plane of simultaneity which is tilting differently based on your direction and the effect of that tilt is enlarged over distance.
– MSH
Commented Jun 28, 2022 at 10:34

The explanation, according to SR, is straightforward, and is exactly analogous to spatial rotations. Imagine a line stretching to infinity from left to right. If you have a second line that is rotated by some tiny angle from the first, the lines will diverge. The further you go from their common origin, the wider the divergence between the lines will be. That is a simple property of geometry.

The same principle applies in spacetime. If you move relative to me, then our respective planes of simultaneity become tilted relative to each other, so the gap between them increases with distance.